# Normal Distributions

## Worksheet 2

Open the interactive version of the normal tables. (Opens a separate browser window.)

Obtain some pictures of normal curves. Print them out and use them to help solve problems.

1. The Graduate Record Examination (GRE) is widely used to help predict the performance of applicants to graduate school. The range of possible scores on a GRE is 200 to 900. The math department at a university finds that the scores of its applicants on the verbal portion of the GRE (VGRE) are approximately normal with mean m = 612 and standard deviation s = 103. If we select an applicant file at random, find

1. The probability VGRE exceeds 800.
2. The probability VGRE is between 400 and 800.
3. The value x such that 10% of applicants score below x.

2. The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 264 days and standard deviation 16 days.

1. What's the probability a pregnancy lasts less that 250 days?
2. What's the probability a pregnancy lasts between 240 and 270 days?
3. Doctors agree that serious thought to labor-inducing procedures ought to accompany any pregnancy that runs in the longest 2% of all pregancies. At what day should these procedures be considered?

3. The rate of return on stock indexes (which combine many individual stocks) is approximately normal. Since the Standard and 1945, Poor's 500 index has had a mean yearly return of about 12%, with a standard deviation of about 16.5%. Assume the normal distribution is the distribution of yearly returns over a long period.

1. In what range do the middle 95% of all yearly returns lie? The middle 68%?
2. The market is down for the year if the return on the index is less than zero. What is the probability the market will be down for a random year?
3. What is the probability the index gains 25% or more in a year?
4. Obtain a 50% prediction interval for a random year's return. (This amounts to finding numbers L and U such that the probability is 0.50 that a yearly gain is between L and U: P[L < x < U] = 0.50.)

## Solutions

Some of your results might be a little different; I've used a better table (more accuracy). If you're real close -- you're right!

1. a) 0.0340, b) 0.9462, c) z = 1.28, so go 1.28 stdevs below the mean: 612 - 1.28(103) = 480.16.

2. a) 0.1908, b) 0.5794, c) = z = 2.05, so go 2.05 stdevs above the mean: 264 + 2.05(16) = 296.8 days.

3. a) -21% to 45%, -4.5% to 28.5%, b) 0.2335, c) 0.2154, d) use z = .674 and z = -.674, go .674 stdevs above and below the mean to arrive at (0.88%, 23.12%).